10 20

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10 19.gif

10_19

10 21.gif

10_21

Contents

10 20.gif
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Knot presentations

Planar diagram presentation X1425 X11,14,12,15 X3,13,4,12 X13,3,14,2 X5,18,6,19 X7,20,8,1 X19,6,20,7 X9,16,10,17 X15,10,16,11 X17,8,18,9
Gauss code -1, 4, -3, 1, -5, 7, -6, 10, -8, 9, -2, 3, -4, 2, -9, 8, -10, 5, -7, 6
Dowker-Thistlethwaite code 4 12 18 20 16 14 2 10 8 6
Conway Notation [352]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif

Length is 12, width is 5,

Braid index is 5

10 20 ML.gif 10 20 AP.gif
[{12, 3}, {4, 2}, {3, 11}, {1, 4}, {10, 12}, {11, 9}, {8, 10}, {9, 5}, {2, 6}, {5, 7}, {6, 8}, {7, 1}]

[edit Notes on presentations of 10 20]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-11][-1]
Hyperbolic Volume 8.31738
A-Polynomial See Data:10 20/A-polynomial

[edit Notes for 10 20's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus 2
Rasmussen s-Invariant -2

[edit Notes for 10 20's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^2+9 t-11+9 t^{-1} -3 t^{-2}
Conway polynomial -3 z^4-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 35, -2 }
Jones polynomial q-1+3 q^{-1} -4 q^{-2} +5 q^{-3} -6 q^{-4} +5 q^{-5} -4 q^{-6} +3 q^{-7} -2 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8+a^8-z^4 a^6-2 z^2 a^6-a^6-z^4 a^4-z^2 a^4-z^4 a^2-2 z^2 a^2-a^2+z^2+2
Kauffman polynomial (db, data sources) z^6 a^{10}-4 z^4 a^{10}+3 z^2 a^{10}+2 z^7 a^9-8 z^5 a^9+7 z^3 a^9-z a^9+2 z^8 a^8-8 z^6 a^8+9 z^4 a^8-5 z^2 a^8+a^8+z^9 a^7-3 z^7 a^7+3 z^5 a^7-4 z^3 a^7+2 z a^7+3 z^8 a^6-12 z^6 a^6+17 z^4 a^6-9 z^2 a^6+a^6+z^9 a^5-4 z^7 a^5+9 z^5 a^5-8 z^3 a^5+3 z a^5+z^8 a^4-2 z^6 a^4+3 z^4 a^4+z^7 a^3-z^5 a^3+2 z^3 a^3-z a^3+z^6 a^2-2 z^2 a^2+a^2+z^5 a-z^3 a-z a+z^4-3 z^2+2
The A2 invariant q^{28}+q^{22}-q^{20}-q^{14}-q^{10}-q^4+2 q^2+1+ q^{-2} + q^{-4}
The G2 invariant q^{142}-q^{140}+2 q^{138}-3 q^{136}+2 q^{134}-2 q^{132}-q^{130}+6 q^{128}-9 q^{126}+10 q^{124}-9 q^{122}+3 q^{120}+4 q^{118}-11 q^{116}+16 q^{114}-13 q^{112}+9 q^{110}+q^{108}-8 q^{106}+12 q^{104}-9 q^{102}+3 q^{100}+3 q^{98}-7 q^{96}+7 q^{94}-2 q^{92}-4 q^{90}+11 q^{88}-12 q^{86}+9 q^{84}-4 q^{82}-5 q^{80}+10 q^{78}-15 q^{76}+15 q^{74}-9 q^{72}+3 q^{70}+5 q^{68}-12 q^{66}+13 q^{64}-10 q^{62}+4 q^{60}+q^{58}-7 q^{56}+7 q^{54}-3 q^{52}-2 q^{50}+5 q^{48}-6 q^{46}+q^{44}+2 q^{42}-7 q^{40}+7 q^{38}-5 q^{36}+3 q^{34}-q^{32}-3 q^{30}+4 q^{28}-5 q^{26}+5 q^{24}-5 q^{22}+4 q^{20}-3 q^{18}-q^{16}+4 q^{14}-6 q^{12}+8 q^{10}-4 q^8+3 q^6+q^4-q^2+4-3 q^{-2} +4 q^{-4} - q^{-6} + q^{-8} + q^{-10} - q^{-12} +2 q^{-14} + q^{-18}